For simplicity, we follow the rules: X denotes a non empty set, x denotes an element of X , f , g denote partial functions from X to R, S denotes a σ-field of subsets of X , F denotes a function from Q into S, p denotes a rational number, r denotes a real number, n, m denote natural numbers, and A, B denote elements of S. Let us consider X and let us consider f . We say that f is finite if and ...

In the paper the existing results concerning a special kind of trajectories and the theory of first return continuous functions connected with them are used to examine some algebraic properties of classes of functions. To that end we define a new class of functions (denoted Conn) contained between the families (widely described in literature) of Darboux Baire 1 functions (DB1) and connectivity ...

Different stationarity and regularity concepts for extended real-valued functions on metric spaces are considered in the paper. The properties are characterized in terms of certain local constants. A classification scheme for stationarity/regularity constants and corresponding concepts is proposed. The relations between different constants are established.

An Open Coloring Axiom type principle is formulated for uncountable cardinals and is shown to be a consequence of the Proper Forcing Axiom. Several applications are found. We also study dense C∗-embedded subspaces of ω∗, showing that there can be such sets of cardinality c and that it is consistent that ω∗ \ {p} is C∗-embedded for some but not all p ∈ ω∗.

The set of interval Hausdorff continuous functions constitutes the largest space preserving basic algebraic and topological structural properties of continuous functions, such as linearity, ring structure, Dedekind order completeness, etc. Spaces of interval functions have important applications not only in the construction of numerical methods and algorithms, but to problems in abstract areas ...

This paper presents a method of inductive inference of real-valued functions from given pairs of observed data of (x; h(x)), where h is a target function to be inferred. Each of such observed data inevitably involves some ranges of errors, and hence it is usually represented by a pair of rational numbers which show the approximate value and the error bound, respectively. On the other hand, a re...

|Associative reinforcement learning (ARL) tasks de ned originally by Barto and Anandan [1] combine elements of problems involving optimization under uncertainty, studied by learning automata theorists, and supervised learning pattern-classi cation. The stochastic real-valued (SRV) unit algorithm [6] has been designed for an extended version of ARL tasks wherein the learning system's outputs can...

We develop a notion of derivative of a real-valued function on a Banach space, called the L-derivative, which is constructed by introducing a generalization of Lipschitz constant of a map. As with the Clarke gradient, the values of the L-derivative of a function are non-empty weak* compact and convex subsets of the dual of the Banach space. The L-derivative, however, is shown to be upper semi c...